On generalized averaged Gaussian formulas. II

Abstract:

Recently, by following the results on characterization of positive quadrature formulae by Peherstorfer, we proposed a new $(2\ell +1)$-point quadrature rule $\widehat G_{2\ell +1}$, referred to as a generalized averaged Gaussian quadrature rule. This rule has $2\ell +1$ nodes and the nodes of the corresponding Gauss rule $G_\ell$ with $\ell$ nodes form a subset. This is similar to the situation for the $(2\ell +1)$-point Gauss-Kronrod rule $H_{2\ell +1}$ associated with $G_\ell$. An attractive feature of $\widehat G_{2\ell +1}$ is that it exists also when $H_{2\ell +1}$ does not. The numerical construction, on the basis of recently proposed effective numerical procedures, of $\widehat G_{2\ell +1}$ is simpler than the construction of $H_{2\ell +1}$. A disadvantage might be that the algebraic degree of precision of $\widehat G_{2\ell +1}$ is $2\ell +2$, while the one of $H_{2\ell +1}$ is $3\ell +1$. Consider a (nonnegative) measure $d\sigma$ with support in the bounded interval $[a,b]$ such that the respective orthogonal polynomials, above a specific index $r$, satisfy a three-term recurrence relation with constant coefficients. For $\ell \ge 2r-1$, we show that $\widehat G_{2\ell +1}$ has algebraic degree of precision at least $3\ell +1$, and therefore it is in fact $H_{2\ell +1}$ associated with $G_\ell$. We derive some interesting equalities for the corresponding orthogonal polynomials.